Alexandr Yurevich Trynin’s research while affiliated with Saratov State University and other places

A mixed boundary value problem with arbitrary continuous, not necessarily satisfying boundary conditions, functions in initial conditions and inhomogeneities of the equation is solved. A method is proposed for finding a generalized solution by a modification of the interpolation operators of functions constructed from solutions of Cauchy problems with second-order differential expression. Methods of finding the Fourier coefficients of auxiliary functions using the Stieltjes integral or the resolvent of the third-order Cauchy differential operator are proposed.

Решена смешанная краевая задача с произвольными непрерывными, необязательно удовлетворяющими граничным условиям, функциями в начальных условиях и неоднородности уравнения. Предложен метод нахождения обобщенного решения с помощью модификации операторов интерполирования функций, построенных с помощью решений задач Коши с дифференциальным выражением второго порядка. Найдены способы нахождения коэффициентов Фурье вспомогательных функций с помощью интеграла Стилтьеса или резольвенты дифференциального оператора Коши третьего порядка. Библиография: 39 наименований.

Let sequences , satisfy the relations , , , as , and let and . We redefine the function as on the interval by polygonal arcs in such a way that the function remains continuous and vanishes on a neighbourhood of the ends of the interval. Also let the function and the pair of sequences , be connected by the equiconvergence condition. Then for the classical Lagrange–Jacobi interpolation processes to approximate uniformly with respect to on it is sufficient that have bounded variation on . In particular, if the sequences and are bounded, then for the classical Lagrange–Jacobi interpolation processes to approximate uniformly with respect to on it is sufficient that the variation of be bounded on , .

Пусть последовательности $\{\alpha_n\}_{n=1}^{\infty}$, $\{\beta_n\}_{n=1}^{\infty}$ удовлетворяют соотношениям $\alpha_n\in\mathbb{R}$, $\beta_n\in\mathbb{R}$, $\alpha_n=o(\sqrt{n/\ln n})$, $\beta_n=o(\sqrt{n/\ln n})$ при $n\to \infty$, а отрезок $[a,b]\subset (0,\pi)$ и функция $f\in C[a,b]$. Доопределим функцию f до F на отрезке $[0,\pi]$ ломаными так, чтобы она, оставаясь непрерывной, исчезала в окрестности концов отрезка $[0,\pi]$. Пусть также функция f и пара последовательностей $\{\alpha_n\}_{n=1}^{\infty}$, $\{\beta_n\}_{n=1}^{\infty}$ связаны между собой условием равносходимости. Тогда для того чтобы классические интерполяционные процессы Лагранжа-Якоби $\mathcal{L}_n^{(\alpha_n,\beta_n)}(F,\cos\theta)$ равномерно по $\theta$ на [a,b] аппроксимировали функцию $f\in C[a,b]$ достаточно ограниченности вариации функции $V^{b}_{a}(f)<\infty$ на отрезке [a,b]. В частности, если последовательности $\{\alpha_n\}_{n=1}^{\infty}$, $\{\beta_n\}_{n=1}^{\infty}$ ограничены, то для того чтобы классические интерполяционные процессы Лагранжа-Якоби $\mathcal{L}_n^{(\alpha_n,\beta_n)}(F,\cos\theta)$ равномерно по $\theta$ на [a,b] аппроксимировали функцию $f\in C[a,b]$ достаточно ограниченности вариации функции, $V^{b}_{a}(f)<\infty$, на отрезке [a,b]. Библиография: 42 наименования.

One functional class is described in terms of one-sided modulus of continuity and the modulus of positive (negative) variation on which thereis a uniform convergence of the truncated cardinal Whittaker functions.

We establish the uniform convergence inside an arbitrary interval (a, b) ⊂ [0, π] for the values of the Lagrange-Sturm-Liouville operators for functions in a class defined by one-side moduli of continuity and oscillations. Outside this interval, the sequence of values of the Lagrange-Sturm-Liouville operators may diverge. The conditions describing this functional class contain a restriction only on the rate and magnitude of the increasing (or decreasing) of the continuous function. Each element of the proposed class can decrease (or, respectively, increase) arbitrarily fast. Popular sets of functions satisfying the Dini-Lipschitz condition or the Krylov criterion are proper subsets of this class, even if, under their conditions, the classical modulus of continuity and the variation are replaced by the one-sided ones. We obtain sharp upper bounds for functions and Lebesgue constants of the Lagrange-Sturm-Liouville processes. We establish sufficient conditions of the uniform convergence of the Lagrange-Sturm-Liouville processes in terms of the maximal absolute value of the sum and the maximal sum of the absolute values of the weighted first order differences. We prove the equiboundedness of the sequence of fundamental functions of Lagrange-Sturm-Liouville processes. Three new operators are proposed, which are modifications of the Lagrange-Sturm-Liouville operator and they allow one to approximate uniformly an arbitrary continuous function vanishing at the ends on the segment [0, π]. All the results of the work remain valid if the one-sided moduli of continuity and oscillations are replaced by the classical ones.

We study approximation properties of various operators being the modifications of sinc approximations of continuous functions on an interval.